Very few scientists dare to doubt conservation of energy, doing so destroys reputation and carrier. Nevertheless, letís take a chance.

"This law of conservation of energy is not something we have derived from basic physics principles. Rather, it is a law based on countless experiments. Scientists and engineers have never found an exception to it. Energy simply cannot magically appear or disappear." (Jearl Walker, David Halliday, Robert Resnick (2014) Fundamentals of Physics, tenth edition, Cleveland State University, Pages 195-196)

Being kids we all played with a gyroscope and you must remember how much harder it is to tilt or flip a rotating toy than a non-rotating one. Now letís think like adults. The only difference between resting and rotating gyroscopes is the rotation. In this particular case, RPM is proportional to kinetic energy. So when we flip it some of that energy must be lost due to resistance to our hands. This is what conservation of energy is about.

Description of the test: Two identical gyroscopes with equal RPM one left along to come to its full stop another we flip and turn over and tumble until it stops (one gyroscope will suffice either). Conservation of energy tells us that the one left along must take more time to stop than the other.

It does not take sophisticated equipment to replicate the test, any kind of electrical gyroscope and a chronometer with one second precision. To get a better idea about the acting forces do it barehanded. Big surprise awaits you.

So when we flip it some of that energy must be lost due to resistance to the force applied by our hands.

I'm not sure that's true , it doesn't necessarily take work to resist a force , only when that force moves through a distance ...

I'm not sure if the rotational speed of a gyroscope changes if it's axxis does .

Consider this gyroscope ....

Gravity is exerting a force on it , and this force causes it to gyrate about it's pivot , it's axis of orientation is constantly changing , but it's speed will remain the same , without friction and wind resistance it will go on forever never slowing or falling to a more horizontal position ... gravity is acting , but no work is done , no speed is lost.